I know the basic premise of tank pressurization. A inert gas (usually Helium) is kept in small tanks in or around the large propellant tanks, and when propellant is burned the helium fills the empty space the burning propellants left. But do both stages need to pressurize the tanks, or only the upper stages? And what's stopping the helium from entering the combustion chamber?
1) "do both stages need to pressurize the tanks, or only the upper stages?"
In general all stages need pressurization. I am unaware of any counterexample. If the engine is pressure-fed, you need higher pressure than the combustion chamber pressure. If the engine is pump-fed, you need sufficiently high pressure to keep the pumps from cavitating.
2) "And what's stopping the helium from entering the combustion chamber?"
Nothing, if you run out of propellant. But for boosters, acceleration holds the heavier propellants in the bottom of the tank, and the lighter helium stays at the top. The situation is more complicated for orbital systems, there are questions on the site about that (like this one: Propellant Settling for RCS Thrusters and this one: Starting a liquid fuel engine in zero gravity)